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Average performance of the sparsest approximation using a dictionary
Prof. Mila Nikolova ( Centre de Mathématiques et de Leurs Applications, ENS Cachan )

2008 - 04 - 11 (Fri.)
10:30 - 11:20
405, Mathematics Research Center Building (ori. New Math. Bldg.)

We consider the minimization of the number of non-zero coefficients (the ℓ0 norm) of the representation of a data set in terms of a dictionary under a fidelity constraint. (Both the dictionary and the norm defining the constraint are arbitrary.) This (non-convex) optimization problem naturally leads to the sparsest representations, compared with other objectives. Our goal is to measure the sets of data yielding a K-sparse or sparser solution - i.e. involving at most K non-zero components. Data are assumed uniformly distributed on a domain defined by a norm. Relevant bounds on the Lebesgue measure of these sets are derived. They naturally lead to bound the probability of getting a K-sparse (or sparser) solution. We specify these results in the case of the Euclidian norm and an arbitrary dictionary. Then we specialize the problem to orthogonal bases (e.g. splines or wavelets) and focus on popular choices for the two norms. We get sharper bounds that are easy to compute. These results have a considerable theoretical and practical importance.

Time:March 7, 2008 - January 31, 2009
Room:
Organizer:I-Liang Chern ( Department of Mathematics, National Taiwan University )

Available Talk List

2008-03-07 (Fri.)		Causality of Wave Propagation (I)
2008-03-14 (Fri.)		Multiscale modeling, asymptotic analysis, and their applications in finance (I,II)
2008-04-11 (Fri.)		Average performance of the sparsest approximation using a dictionary
2008-04-25 (Fri.)		Anomalous Dispersion and Superluminal Propagation
2008-05-09 (Fri.)		Imaging of Multiple Targets in Rician/Rayleigh Fading Media
2008-05-09 (Fri.)		What are networks？
2008-05-16 (Fri.)		Some Effective Properties of the Periodic Structure: a Homogenization Theory Study
2008-05-23 (Fri.)		生物運動力學的若干研究
2008-09-26 (Fri.)		Introduction to Homogenization Theory I, II
2008-10-03 (Fri.)		Two-Scale Convergence and Homogenization of Periodic Structures
2008-10-17 (Fri.)		Homogenization Theory for the Elasticity I, II
2008-10-31 (Fri.)		Memory Effects and Homogenization
2008-11-14 (Fri.)		Maxmizing Band Gaps in 2-D Photonic Crystals
2008-11-21 (Fri.)		What is weak convergence?
2008-11-27 (Thu.)		What is weak convergence?
2008-12-26 (Fri.)		Solving Guided Wave Modes in Plasmonic Crystals
2009-01-09 (Fri.)		Solving Guided Wave Modes in Plasmonic Crystals II
2009-01-16 (Fri.)		Band Gap Optimization for Photonic Crystals -Part I: Introduction
2009-02-20 (Fri.)		Investigation of Transport and Mixing in Microchannel Driven by Capillary Pumping
2009-02-27 (Fri.)		Band Gap Optimization for Photonic Crystals (Part II: Result and Discussion)
2009-03-06 (Fri.)		Analytical solution of electro-osmotic flow in a periodic structure of the filter
2009-03-13 (Fri.)		Homogenization Theory and Its Application
2009-03-20 (Fri.)		Review of achieving transparency with plasmonic and metamaterial coatings
2009-05-01 (Fri.)		Revisiting the aerodynamics of hovering flight using simple models
2009-05-22 (Fri.)		Electro-osmotic flow in a micro-channel with corrugated walls:coherence between the electric potential and the wall shape function
2012-12-05 (Wed.)		SSS